Example of a weighted algebra \(\mathcal L_p^w(G)\) on an uncountable discrete group
Let \(G\) be a locally compact group. The author considers the question when a weighted \(L_p\)-space on \(G\) (denoted here by \(L_p^w(G)\)) is a Banach algebra under the convolution multiplication. She shows that for any discrete free group such a weight exists, provided \(1\leq p\leq 2\) (so that \(G\) need not be \(\sigma\)-compact). If the group in question is uncountable, then no such weight can exist, if \(p>2\). She also shows that if \(G\) is amenable and \(L^w_p(G), p>1\), is an algebra for some weight, then \(G\) is \(\sigma\)-compact.
scientific article; zbMATH DE number 5553645